This is very similar to conventional air/water/salt absorbtion chilling, with one important difference (which I'll describe when I get to it).

There are three "loops" of fluids involved in this system: air, liquid desiccant (something cheap and safe, like calcium chloride), and distilled water.

The
first loop is air:

Fresh air moves from the interior of the home, through three handlers, then back into the home. In the first and third air handlers, air and concentrated liquid desiccant meet and part, making the air dryer and the desiccant more dilute. In the second air handler, distilled water and air meet and combine, making the air cooler (and more humid), and the water disappear (into the air).

The temperature of the concentrated desiccant and of the distilled water, before it reaches the air, is approximately that of the atmosphere outside the home. The net effect is to make the air cooler and dryer.

The second loop is the liquid desiccant:

[edited for simplified design]

The dilute desiccant, after having dehumidified the air, enters a well insulated boiler. Inside of that boiler are three very long (but coiled) tubes of heat providing fluid, which are close (to fill as much of the boiler's volume as possible) but not touching each other (so that the heat from within the tubes will move into the dilute desiccant, not into the other tubes).

The first tube heating the contents of the boiler is filled with the concentrated liquid desiccant. The top of the tube is several inches below the top of the boiler, and acts much like an overflow tube.

The second tube heating the contents of the boiler is filled with compressed steam (in the upper portion) and hot water (still under pressure) in the lower portion. After the pure water leaves the boiler, it passes through an expansion valve. Since the pure water should be well below 100C by this point, it won't re-boil.

The third tube contains hot fluid, either flue gasses from oil or gas combustion, or hot hydronic fluid from a solar collector.

For all three tubes, the contents of the tube enter at the top, and leave at the bottom.

The concentrated desiccant coming out of the boiler cools further in a cooling tower.

Then, it goes back to the air handlers bringing it in contact with household air.

The third loop is water:

The steam from the boiler goes to an electrically powered gas compressor, then, under pressure, into a condensor, which is a spiral tube inside of the boiler. After condensing and cooling inside the boiler, the condensate passes through an expansion valve.

This water will then pass through a cooling tower, then re-enter the home, to be sprayed into the air between the two liquid desiccant air handlers.

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What's the advantage of this over conventional air conditioning?

The first thing that comes to mind is that the refrigerant the system uses is water (aka R718), which is the safest substance one can image for the task.

Unlike other R718 systems, the compressor can be very small, since it raises the pressure starting from atmospheric pressure, instead of raising pressure starting from a partial vacuum.

Furthermore, the compressor only needs enough of a pressure / temperature increase to boil the dilute desiccant (which has already been preheated by heat exchangers)... and the lower the temperature lift, the higher the COP of the system.

Another advantage to this design: Since vapor compression distillation requires both mechanical (electrical) power (for compression), and thermal (gas/oil) power to work, we can optomize how much of each the machine uses, depending on the cost of electricity (which some electricity providers vary, depending on time of day).

If electricity is cheaper ("off peak" power), we compress the steam to a higher pressure, which moves more heat mechanically (electrically), which means that less thermal energy (gas/oil) is needed. When electricity is costlier, a lower compression ratio is used, which requires more fuel be consumed in it's place.